Log-concavity for independent sets of valuated matroids

TL;DR

This paper extends the proof of the Mason-Welsh conjecture to valuated matroids and M-convex functions using Lorentzian polynomials, demonstrating log-concavity in these generalized settings.

Contribution

It introduces a novel approach using Lorentzian polynomials to prove log-concavity for valuated matroids and related structures, generalizing previous results.

Findings

Proves log-concavity for valuated matroids and M-convex functions.

Extends Mason-Welsh conjecture to valuated discrete polymatroids.

Provides log-concavity results for valuated bimatroids.

Abstract

Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials, developed by Br\"and\'en and Huh, to prove a generalization of the Mason-Welsh conjecture to the context of valuated matroids. In fact, we provide a log-concavity result in the more general setting of valuated discrete polymatroids, or equivalently, M-convex functions. Our approach is via the construction of a generic extension of a valuated matroid or M-convex function, so that the bases of the extension are related to the independent sets of the original matroid. We also provide a similar log-concavity result for valuated bimatroids, which, we believe, might be of independent interest.